Automatic Train Stop Control System

ABSTRACT

A method controls a movement of a train to a stop at a stopping position between a first position and a second position. The method determines constraints of a velocity of the train with respect to a position of the train forming a feasible area for a state of the train during the movement, such that an upper curve bounding the feasible area has a zero velocity only at the second position, and a lower curve bounding the feasible region has a zero velocity only at the first position. Next, the method controls the movement of the train subject to the constraints.

FIELD OF THE INVENTION

The present invention relates generally to automated process control,and more particularly to a system and a method for stopping a train at aposition with an automatic control.

BACKGROUND OF THE INVENTION

The Train Automatic Stop Control (TASC) system, which is often part ofan Automatic Train Operation (ATO) system, manages the train brakingsystem to stop the train at the predetermined location. The TASC systemreceives measurements from sensors, on the train and/or on remotestations via communication networks, estimates the state of the trainincluding a position and a velocity of the train, and selects theactions for the braking system. These steps are repeated multiple timesuntil the train stops.

The TASC system allows the trains equipped with TASC to stopautomatically at stations without the need to operate the brakesmanually. The TASC was originally developed in the 1950s and the 1960sas a way of ensuring that trains stop properly at stations, especiallyif the driver has made a minor driving lapse and stopped with a slightoverrun/underrun. When station platforms are provided with screen doors,the doors of the train must be aligned with the platform doors asotherwise the operation of automatic trains, particularly driverlessunderground trains, is disrupted.

Most of the conventional methods select the control action in the TASCsystem according to one of many possible velocity profiles determinedbased on a distance between the current position of the train and thestop position see, for example, U.S. 2013/0151107. The velocity profileis called a run curve. If a distance along the route is denoted by z,then a desired velocity v(z) at position z describes the run curve. Therun curve has to obey legal and mechanical constraints of the route,e.g. speed limits, safety margins, and must be physically realizable bymechanisms of the train.

However, the generation of those velocity profiles are difficult and/ortime and resource consuming. In addition, the selection of the optimalvelocity profile is prone to errors due to uncertainty of some of theparameters of the movement of the train, such mass of the train mass,and the track friction. In practice, many reference profiles aregenerated before train operation based on different assumptions of trainand environmental parameters, and the one to be used in each operationsof stopping is selected based on evaluating the current conditions.However, there is not guarantee that one curve satisfying exactly thecurrent conditions is available, and/or that the current conditions areexactly known, and/or that the conditions do not change during executionof the stopping.

For instance, a run curve can be selected based on high friction of therails as in the case of dry rails to minimize the stopping time byexploiting the high rail friction. If the rail conditions change duringthe stopping, for instance due to encountering a section of track wherethe rails are wet which reduces the rail friction, it may be impossibleto achieve the desired braking effort. Hence the train velocity profilewould deviate from the run curve, and the train stop pasts its desiredstopping point, missing alignment with the station.

Furthermore, separation of trajectory generation and control to followtrajectory can fail to follow the selected run curve exactly due to,e.g., the imprecisions of the braking system, change of the trainparameters, and external disturbances, so the train can fail to stop atdesired location. In theory, a feedback control can aim to track theselected run curve while reducing the effect of external uncertaintiesand improperly selected assumptions. However, the feedback controlusually cannot provide definite guarantees of the performance intracking an externally generated signal.

Furthermore, it is in general not optimal to first generate a trajectoryand then control the train to follow the trajectory based on feedbackfrom sensors that adjusts to current conditions, due to a two stepsdesign procedure. In addition, the concurrent generation of thetrajectory and feedback control action subject to uncertainty in theparameters it is notoriously difficult to achieve because theuncertainty reduces the accuracy of prediction of the future behavior ofthe train, which is required for optimization.

Accordingly, there is a need to provide a system and a method forstopping a train at a position with an automatic control, but withoutthe predetermined velocity profiles.

SUMMARY OF THE INVENTION

Some embodiments of the invention are based on the realization that itis possible to define constraints of a state of a movement of the train,such that a constraint movement of the train stops of the train at apredetermined stopping range. Thus, the control of the movement of thetrain according to a velocity pattern that is prone to errors can besubstituted with a control of the movement of the train subject to thosespecial constraints which are independent of the train and environmentparameters.

Specifically, some embodiments of the invention determines a feasibleregion for a state of the train, such that the feasible region includesat least one state with zero velocity at the stopping range, andcontrols the movement of the train such that the state of the train isalways within that region. For example, some embodiments design acontroller those select the braking system action that to maintain thestate of the train within the feasible region by repeatedly solving anoptimization problem. Accordingly, some embodiments of the inventiontransform the tracking problem into an optimization problem subject toconstraints.

In such a manner, the embodiments enable stopping a train at a positionwith an automatic control, but without the predetermined velocityprofiles. This is because the constraints on the state of the movementof the train that guarantees the stopping of the train at thepredetermined stopping range can be generated without the velocityprofiles. For example, instead of generating multiple velocity profiles,only two constraints defining a lower and an upper curve of the feasibleregion can be determined. It is also realized in this invention that theselection of the constraints affects the minimum and maximum arrivaltime of the train at the position, such that the time of arrival can beused as guidance for generating those constraints.

For example, some embodiments determine a lower curve and an upper curvebounding a velocity of the train with respect to a position of thetrain, such that the upper curve has a zero velocity only at thefarthest border of a stopping range, and the lower curve has a zerovelocity only at the nearest border of the stopping range, and determinethe feasible region for a state of the train using the lower and theupper curves and mechanical and/or legal constraints on the movement ofthe train.

For example, in one embodiment the upper curve is a first line with afirst slope, and the lower curve is a second line with a second slope.Usually, the first slope is greater than the second slope to enforce asufficient size for the feasible region. This embodiment reduces theselection of the constraints only to the values for the slopes of thefirst and the second lines.

It is also realized that the selection of the constraints affects theminimum and maximum arrival time of the train at the stopping range, andthe desired arrival time can be used in the selection of the twoparameters. For example, one embodiment selects the value of the firstslope based on a minimal stopping time, and selects the value of thesecond slope based on a maximal stopping time.

Due to the nature of optimization-based receding horizon control, theexistence of a solution for a certain horizon does not by itselfguarantees the existence of the solution for a subsequent horizon.However, some embodiments of the invention are based on yet anotherrealization that it is possible to select a subset of the feasibleregion, such that from any state of the train and any possiblevariations in the parameters of the movement of the train, there is acontrol maintaining the state of the train within the subset.Accordingly, if a cost function representing the movement of the trainis optimized subject to constraints defined by that special subset ofthe feasible region, as contrasted with the optimization within thefeasible region itself, there is a guarantee that the train stops withinthe predetermined stopping range.

Accordingly, one embodiment discloses a method for controlling of amovement of a train to a stop at a stopping position between a firstposition and a second position. The method includes determiningconstraints of a velocity of the train with respect to a position of thetrain forming a feasible area for a state of the train during themovement, such that an upper curve bounding the feasible area has a zerovelocity only at the second position, and a lower curve bounding thefeasible region has a zero velocity only at the first position; andcontrolling the movement of the train subject to the constraints. Thesteps of the method are performed by a processor.

Another embodiment discloses a system for controlling of a movement of atrain to a stop at a stopping position between a first position and asecond position. The system includes a constraint generating unit fordetermining constraints of a velocity of the train with respect to aposition of the train forming a feasible area for a state of the trainduring the movement, such that an upper curve bounding the feasible areahas a zero velocity only at the second position, and a lower curvebounding the feasible region has a zero velocity only at the firstposition, wherein the upper curve is a first line with a first slope,and the lower curve is a second line with a second slope and the firstslope is greater than the second slope; and a controller for controllingthe movement of the train subject to the constraints.

Yet another embodiment discloses a method for controlling of a movementof a train to a stop at a stopping position between a first position anda second position. The method includes determining constraints of avelocity of the train with respect to a position of the train forming afeasible area for a state of the train during the movement, such that anupper curve bounding the feasible area has a zero velocity only at thesecond position, and a lower curve bounding the feasible region has azero velocity only at the first position; determining a controlinvariant subset of the feasible region, wherein for each state withinthe control invariant subset there is at least one control actionmaintaining the state of the train within the control invariant subset;and controlling the movement of the train subject to the constraints byselecting a control action maintaining the state of the train within thecontrol invariant subset of the feasible region. The steps of the methodare performed by a processor.

BRIEF DESCRIPTION OF THE FIGURES

FIGS. 1A and 1B are schematics of a system for controlling of a movementof a train to a stop at a stopping position according to one embodimentof the invention;

FIG. 2A is an example of the feasible area for a state of the trainduring the movement determined according some principles of embodimentof the invention;

FIG. 2B is a block diagram of a method for controlling of a movement ofa train to a stop at a stopping position according to one embodiment ofthe invention;

FIG. 3 is an example of the feasible region for the movement of thetrain defined by linear stopping constraints according to one embodimentof the invention;

FIGS. 4 and 5 are plots for selecting the parameters of the stoppingconstraints based on desired time of arrival of the train according toone embodiment of the invention;

FIG. 6 is a block diagram of a method for predictive constrained controlaccording one embodiment of the invention;

FIG. 7A is an example control invariant subset of the feasible regionaccording one embodiment of the invention;

FIG. 7B is a schematic of the relation between different feasibleregions according one embodiment of the invention;

FIG. 8 is a block diagram of a backward-reachable region computation fordetermining the control invariant subset starting from the feasibleregion according one embodiment of the invention;

FIG. 9 is a block diagram of an exemplar implementation of determiningthe previous set of states according one embodiment of the invention;

FIG. 10 is a block diagram of a method for computation of the couples ofstates-inputs according one embodiment of the invention;

FIG. 11 is a diagram of a predictive control system according toaccording one embodiment of the invention; and

FIG. 12 is a block diagram of a controller that does not use a fullmodel of the train dynamics and constraints according to an alternativeembodiment of the invention.

DETAILED DESCRIPTION

FIGS. 1A and 1B show a system for controlling of a movement of a train110 to a stop at a stopping position. In this disclosure, the term“train” is used generally and includes any guided means oftransportation, such as, but not limited to, electrical trains, guidedtransport systems at airports, or guided hybrid cars. The train can beprovided with wheels, often referred to as rolling stocks that are incontact with and roll on the rail tracks.

A control system 101 controls the movement of the train 110 travelingtowards a station 2 such that the train stops at a pre-determined rangeof positions 3 between a first position 7 and a second position 6without stopping anywhere else. Some embodiments select a referencesystem 100 having the origin 5 at a point 4 in the stopping range 3.Thus, the first positions ε_(min)<0, and the second position ε_(max)>0,ε_(max)>ε_(min) are the nearest 7 and furthest 6 positions with respectto the current position of the train where the train is allowed to stop.As used herein, when the train moved to the stop at the stoppingposition, the train has a zero velocity at the stopping position.

The current position d 10 of the train can be determined as the distanceof a specific point 8 of the train, such as the center of the first door9, from the origin 5 of the reference system, where d is negative whenthe train is at a position before the origin with respect to the normaldirection of movement of the train. The velocity of the train 11 is v,where v is positive when the train is moving in its normal direction ofthe movement.

A control system 101 of the train 110 can include one or combination ofa constraints generating unit 1, a control invariant subset generatingunit 3, a train control device 5, and a control computer 7. In someembodiments, the constraints generating unit 1 determines stoppingconstraints 111 of a velocity of the train with respect to a position ofthe train forming a feasible area for a state of the train during themovement leading the train to the stop, and the control computer 7controls the movement of the train subject to the constraints. Thecontrol can be achieved, e.g., by generating a control input 117 to thetrain control device 5 controlling 115 the break system of the train.

In various embodiments, the stopping constraints are determined withouthaving a predetermined run-curve leading the train from the currentposition to the stopping position. In effect, the control of themovement of the train according to a velocity pattern that is prone toerrors is substituted with a control of the movement of the trainsubject to those special constraints. Accordingly, some embodiments ofthe invention transform the tracking problem into an optimizationproblem subject to constraints. Such transformation is advantageous,because the constrained control can guarantee that the constraints arealways satisfied. That guarantee might not be possible for the trackingaccording to a predetermined run-curve.

For example, some embodiments determine, for each time step of control,a control action moving the train from a current position to a nextposition within the feasible region. In those embodiments, thecontrolling includes determining a sequence of control inputs forming anad-hoc run-curve leading the train from the current position to thestopping position. Such ad-hoc run-curve determination is advantageousbecause eliminates efforts need to generate and test predeterminedrun-curves. Also, reformulating the stopping into a constrained problemallows handling the stopping constraints with other constraints on themovement of the trains, such as constraints on traction and brakingforce range, actuator rate, and/or maximum and minimum speed of thetrain.

However, due to the nature of optimization-based receding horizoncontrol, the existence of a solution for a certain horizon does not byitself guarantees the existence of the solution for a subsequenthorizon. Thus, some embodiments also optionally include the controlinvariant subset generating unit 3 for selecting a control invariantsubset 113 from the feasible region defined by the stopping constraints.These embodiments are based on yet another realization that it ispossible to select a subset of the feasible region, such that from anystate of the train and any possible variations in the parameters of themovement of the train, there is a control maintaining the state of thetrain within the subset.

For example, some embodiments design a controller those select thebraking system action that to maintain the state of the train within thefeasible region by repeatedly solving an optimization problem.Accordingly, if a cost function representing the movement of the trainis optimized subject to constraints defined by that special controlinvariant subset of the feasible region, as contrasted with theoptimization within the feasible region itself, there is a guaranteethat the train stops within the predetermined stopping range. Forexample, in various embodiments, the cost function represents one orcombination of the energy consumption of the train during the trip, atime of the trip, both the energy consumption and the time of the trip,or the energy consumption for a predetermined time of the trip.

Soft Landing Constraints

For example, to stop the train at the stopping position within thestopping range, it is sufficient for the train distance from target d,and velocity v, to satisfy at any time instant soft landing constraints

v(t)≦Γ_(max)(ε_(max) −d(t))

v(t)≧Γ_(min)(ε_(min) −d(t)),  (1)

wherein Γ_(max)(s), Γ_(min)(s) are the upper border function and thelower border function that are defined in the range sε(−∞,c] wherec≧ε_(max), are continuous, greater than 0 when their arguments arepositive, smaller than 0 when their arguments are negative, and 0 whentheir arguments are 0. Furthermore for any sε(−∞, c],Γ_(max)(s)≧Γ_(min)(s) and Γ_(max)(c)=Γ_(min)(c).

FIG. 2A shows an example of the feasible area 15 for a state of thetrain during the movement. In this example, a Cartesian axis with trainposition d is on the x axis 20, and train velocity v is on the y axis21. An upper curve 22 bounding the feasible area 15 has a zero velocityonly at the second position 12, i.e., the upper curve intersects the xaxis at a distance 12 equal to ε_(max). Similarly, a lower curve 23bounding the feasible region has a zero velocity only at the firstposition 13, i.e., the lower curve intersects the x axis at a distance13 equal to ε_(min). The upper and the lower curves can intersect at thesame point 14 at distance c>ε_(max).

Intuitively, if the feasible area 15 includes the current position ofthe train and the state of the train is controlled to be maintainedwithin the feasible area 15, at some instant of time the state of thetrain is guaranteed to be on a segment 16 between the points 13 and 12,which corresponds to a zero velocity of the train at the predeterminedstopping range.

For example, when d<ε_(min) the constraints (1) forces the trainvelocity to be positive, so that the train moves towards the target,when d>ε_(max) the constraints (1) forces the train velocity to benegative and hence the train moves towards the target, and that henceany trajectory of the train must include a point of zero velocity in therange of positions between ε_(max) and ε_(min), which means that thetrain stops at a desired stopped range.

FIG. 2B shows a block diagram of a method for controlling of a movementof a train to a stop at a stopping position between a first position 255and a second position 250 according to one embodiment of the invention.The method determines 230 constraints 235 of a velocity of the trainwith respect to a position of the train. The constraints 235 arestopping constraints that form a feasible area for a state of the trainduring the movement, such that satisfaction of the stopping constraintguaranties the stopping of the train at the stopping position. Theconstraints 235 are determined such that an upper curve bounding thefeasible area has a zero velocity only at the second position, and alower curve bounding the feasible region has a zero velocity only at thefirst position. Next, the method controls the movement of the trainsubject to the constraints. Steps of the method are performed by aprocessor 251.

FIG. 3 shows an example of the feasible region 101 for the movement ofthe train defined by linear stopping constraints according to oneembodiment of the invention. In this embodiment, the upper curve is afirst line 102 with a first slope 103, and the lower curve is a secondline 105 with a second slope 106 and the first slope is greater than thesecond slope. This embodiment simplifies the selection of the stoppingconstraints in determining the feasible region for stopping the train.

For example, the constraints can be written in a linear form accordingto

v(t)≦γ_(max)(ε_(max) −d(t))

v(t)≧γ_(min)(ε_(min) −d(t)),  (2)

wherein γ_(max), γ_(min) are two coefficients where γ_(min)>0γ_(max)>γ_(min). If the constraints in (2) are satisfied at any timeinstants, then the train stops between ε_(max) and ε_(min).

A cone-shaped a region 101 in the space of train positions 110 and trainvelocities 120 is referred herein as a soft landing cone. The region 101is delimited by two lines, each corresponding to one of the equations in(2), satisfied with equality. The upper border 102 of the soft landingcone is defined by γ_(max) and ε_(x) where γ_(max) determines the slope103 and ε_(max) determines the intersect 104 of the upper border withthe line of 0 velocity. Similarly the lower border 105 of the softlanding cone is defined by γ_(min) and ε_(min) where γ_(min) determinesthe slope 106 and ε_(min) determines the intersect of 107 the lowerborder with the line of zero velocity.

If the train positions and velocities remain in the soft landing conethe train stops at the stopping range. The parameters ε_(max) andε_(min) define the desired stopping range, because the train stops inthe area 108 between positions ε_(max) and ε_(min) including thestopping position 109 with d=0.

In addition some variations of this embodiment determine the parametersγ_(max) and γ_(min) using the desired timing to stop. For example, theembodiment can select the first slope 103 based on a minimal stoppingtime, and select the second slope 106 based on a maximal stopping time.

FIGS. 4 and 5 show plots for selecting the parameters of the stoppingconstraints based on desired time of arrival of the train. As shown inFIG. 4, for given γ_(max), γ_(min), and initial position 201 d(0), forany ρ>0 defining a tolerance to the acceptable range of stop positions210, the line d=ε_(min)−ρ, 211, in the soft landing cone is reached inthe time interval

$\begin{matrix}{\hat{t} \in \left\lbrack {{\frac{1}{\gamma_{\max}}{\log \left( \frac{ɛ_{\max} - {d(0)}}{ɛ_{\max} - ɛ_{\min} + \rho} \right)}},{\frac{1}{\gamma_{\min}}{\log \left( \frac{ɛ_{\min} - {d(0)}}{\rho} \right)}}} \right\rbrack} & (3)\end{matrix}$

wherein the upper and lower bounds corresponds to corresponds to thetime of the sequence of positions and velocities described by a line 203for the upper bound and by a line 202 for the lower bound.

Similarly, as shown in FIG. 5, for given parameters γ_(max), γ_(min),and initial position 301 d(0), for any σ>0 defining a tolerance to thestop velocity 310, the line v=σ 311 in the soft landing cone is reachedat

$\begin{matrix}{{{\hat{t}(\sigma)} \in \left\lbrack {{\frac{1}{\gamma_{\max}}{\log\left( \frac{\gamma_{\max}\left( {ɛ_{\max} - {d(0)}} \right)}{{\gamma_{\max}\left( {ɛ_{\max} - ɛ_{\min}} \right)} + {\sigma \frac{\gamma_{\max}}{\gamma_{\min}}}} \right)}},{{\frac{1}{\gamma_{\min}}{\log \left( \frac{\gamma_{\min}\left( {ɛ_{\min} - {d(0)}} \right)}{\sigma} \right)}} + \frac{ɛ_{\max} - ɛ_{\min}}{\sigma} + \left( {\frac{1}{\gamma_{\min}} - \frac{1}{\gamma_{\max}}} \right)}} \right\rbrack},} & (4)\end{matrix}$

which corresponds to the sequence of positions and velocities describedby a line 303 for the upper bound and a line 302 for the lower bound.

Reducing a value of the parameter γ_(min) increases the maximum time toreach the stopping position. Increasing a value of the parameter γ_(max)decreases the minimum time to reach the stop. Also, taking γ_(max) andγ_(min) with closer values reduces the difference between minimum andmaximum time to stop, while on the other hand reduces the area of thesoft landing cone which amounts to reducing the number of possible traintrajectories in such a cone.

Constrained Control

Constrained control of the train that enforces the constraints in (1)guarantees that the train stops in the stopping range. However, thetrain position and velocity depends on the actual train dynamicsgenerated by actuating the traction and braking system of the train.Thus, some embodiments of the invention determines a control system toactuate the train traction and braking system so that the train dynamicssatisfies the constraints in (1).

The train dynamics can be described by

{dot over (x)}(t)=f(x(t),u(t),p)

y(t)=h(x(t))  (5)

where x is the train state, u is the train input, p are the trainparameters, y=[d v] is the output vector, f describes the variation ofthe state as a function of the current state, current input and currentparameters, and h describes the output as a function of the currentstate, only.

The state and input variables in (5) are subject to the constraints

xε

,  (6)

uε

,  (7)

where (6) define a set of admissible values for the state variables, and(7) a set of admissible values for the input variables in (5).

In one embodiment of this invention, for a train provided with rollingstocks (wheels) the train dynamics (5) is described by an affine modelobtained by considering a velocity-affine model for the resistance forceto motion,

F _(res)(t)=−c ₀ μg−c ₁ v(t),  (8)

where c₀ is the coefficient of the constant term which models rollingresistance, and c₁ is the coefficient of the linear term which modelsbearing friction and air resistance at low speeds, μ is the frictioncoefficient between the rails and the rolling stocks, g is the gravityacceleration constant. In this embodiment the train dynamics isdescribed by

$\begin{matrix}{{{\overset{.}{d}(t)} = {v(t)}},{{\overset{.}{v}(t)} = {{\frac{k_{a}}{rm}{\chi (t)}} - \frac{c_{0}\mu \; g}{m} - {\frac{c_{1}}{m}{v(t)}}}},{{\overset{.}{\chi}(t)} = {{{- \frac{1}{\tau_{a}}}{\chi (t)}} + {\frac{1}{\tau_{a}}{u(t)}}}},} & (9)\end{matrix}$

where in is the train mass, r is the radius of the wheels, k_(a) is themaximum force, τ_(a) is the actuator time constant.

The affine model of the train dynamics is

{dot over (x)}(t)=A(p)x(t)+B(p)u(t)+B _(w) w(p)  (10)

where the state is x=[dvχ]′ the input u is the command to the forcegenerating actuators from traction (when positive) and braking (whennegative), is the constant resistance term obtained from (9) and thematrices A(p), B(p), are obtained also from (9), where the vector ofparameters p include the train mass, the friction coefficient, thegravity acceleration constant, the maximum force, the actuator timeconstant. In model (10)

$\begin{matrix}{{B_{w} = \begin{bmatrix}0 \\{- 1} \\0\end{bmatrix}},{{w(p)} = {\frac{c_{0}\mu \; g}{m}.}}} & (11)\end{matrix}$

In other embodiments of this invention similar models can include othereffects, for instance the effects of the railroad grade. Thus, the traincontrol system selects the values for the train input function u thatgenerates admissible solution for

$\begin{matrix}{{{\overset{.}{x}(t)} = {f\left( {{x(t)},{u(t)},p} \right)}}{{y(t)} = {\begin{bmatrix}{d(t)} \\{v(t)}\end{bmatrix} = {h\left( {x(t)} \right)}}}{{v(t)} \leq {\Gamma_{\max}\left( {ɛ_{\max} - {d(t)}} \right)}}{{v(t)} \geq {\Gamma_{\min}\left( {ɛ_{\min} - {d(t)}} \right)}}{{{x(t)} \in \chi},{{u(t)} \in },}} & (12)\end{matrix}$

where the sets

,

describe admissible values for the state and input (e.g., maximum andminimum velocity, maximum and minimum actuator positions, etc), and thesolution is sought form current time T for all times in the future(i.e., [T, t_(f)], where t_(f)=∞).

For instance, the sets X, U can model ranges on allowed velocity, therange of actuators, the range of torques obtained by the braking andtraction system, and other specific constraints that describe thedesired operation of the train system.

For instance, the constraint

{dot over (v)}≦0,

which imposes that the train constantly decelerates, i.e., no increasein velocity is allowed, or its relaxed form{dot over (v)}≦ψ(−d),where ψ is a nonnegative, monotonically decreasing function, while d<0relaxes the previous constraints by allowing greater acceleration whenthe train is closer to the stopping position, to improve accuracy of thecontrol.

Some embodiments of this invention optimizing the movement of the trainfrom the current state to subsequent states, and determine a solution to(12) by solving the constrained optimal control problem

$\begin{matrix}{{{\min \mspace{11mu} {F\left( {x\left( t_{f} \right)} \right)}} + {\int_{t_{0}}^{t_{f}}{{L\left( {{x(t)},{u(t)}} \right)}{t}}}}{{\overset{.}{x}(t)} = {f\left( {{x(t)},{u(t)},p} \right)}}{{y(t)} = {\begin{bmatrix}{d(t)} \\{v(t)}\end{bmatrix} = {h\left( {x(t)} \right)}}}{{v(t)} \leq {\Gamma_{\max}\left( {ɛ_{\max} - {d(t)}} \right)}}{{v(t)} \geq {\Gamma_{\min}\left( {ɛ_{\min} - {d(t)}} \right)}}{{{x(t)} \in \chi},{{u(t)} \in }}{{{x\left( t_{0} \right)} = x_{0}},}} & (13)\end{matrix}$

where t₀ is the initial time, x₀ is the state at the initial time, F isthe terminal cost function and L is the stage cost function. If theproblem in (13) can be solved for final time t_(f)=∞, then the stoppingconstraints are always satisfied and the train stops where required.

However, the problem in described in Equations (12) and (13) require thecomputation of an infinitely long control signal for a system subject toan infinite number of constraints are difficult to solve in the traincontrol system directly. Thus, some embodiments solve the problem indescribed in Equations (12) and (13) in a receding horizon fashion.

FIG. 6 shows a block diagram of a method for predictive constrainedcontrol according one embodiment of the invention. The method determines501 a current state x(T) of the movement of the train at a certain timeT, and the movement of the train is optimized 503 from the current stateto subsequent states set t₀=T, t_(f)=T+h 502 over a finite horizon oftime. The optimization 503 solves the constrained optimization problemsubject to the stopping constraints to produce a sequence of controlinputs for the horizon of time h.

The method selects and applies 504 a first control input from thesequence of control inputs specifying the control action for a next timestep of control. For example, the finite horizon control input signal uis applied during the time interval [T, T+dh] 504. Then 505, at timeT+dh, where dh<h a new problem is solved with t₀=T+dh, t_(f)=T+dh+h andthe newly computed input signal is applied, and the steps of the methodare iteratively repeated.

Control Invariant Subset

FIG. 7A shows an example control invariant subset of the feasible regionand selection of a control action maintaining a state of the trainwithin a control invariant subset. According to various embodiments ofthe invention, for each state within the control invariant subset thereis at least one control action maintaining the state of the train withinthe control invariant subset.

Due to the nature of receding horizon control, the existence of asolution for a certain horizon does not by itself guarantees theexistence of the solution for a subsequent horizon. Specifically, whilethe receding horizon solution makes the problem in (13) computationallyfeasible, it is not possible to guarantee that such problem always has asolution. In particular it is possible that the problem in (13) solvedat time T has a solution, but the one to be solved at time T+dh doesnot. This is due to the fact that as the horizon is shifted, theconstraints in (2), (6), (7) have to be enforced on a new piece of thetrajectory, i.e., during the time interval [T+h, T+dh+h] that was notaccount for before.

For example, the state of the machine and a state of the train 520 canbe optimal and feasible for one iteration, but all control actions521-523 that controller is allowed to take during the next iteration canbring a state of the train outside of the feasible region 101.

Some embodiments of the invention are based on yet another realizationthat it is possible to select a subset 401 of the feasible region 101,such that from any state of the train within that subset, there is acontrol action maintaining the state of the train within the subset. Forexample, for any state such as a state 530 within the subset 401 andwithin all possible control actions 531-534 that the controller canexecute, there is at least one control action, e.g., actions 531 and532, that maintains the state of the train within the control invariantsubset 410.

Accordingly, if a control action for controlling the operation isselected such that the state of the train remains in that special subset401 of the feasible region, and the feasible region is generated alsoaccording to Equation (1), then there is a guarantee that it is possibleto determine the sequence of control actions forming an ad-hoc run-curveleading the train from the current position to the stopping position.

For example, one embodiment determines a discretized version of theproblem in (13) by considering a sampling period dh and obtaining adiscrete time model for the dynamics in (5) which is

x(t+dh)=f _(d)(x(t),u(t),p)

y(t)=h _(d)(x(t)),  (14)

wherein given a state x and a control input u, f_(d)(x,u,p) is theupdated state.

Based on the discrete time model, the constrained control is

$\begin{matrix}{{{\min \mspace{11mu} {F\left( {x(N)} \right)}} + {\sum\limits_{k = 0}^{N}{L\left( {{x(k)},{u(k)}} \right)}}}{{x\left( {k + 1} \right)} = {f_{d}\left( {{x(k)},{u(k)},p} \right)}}{{y(k)} = {\begin{bmatrix}{d(k)} \\{v(k)}\end{bmatrix} = {h_{d}\left( {x(k)} \right)}}}{{v(k)} \leq {\Gamma_{\max}\left( {ɛ_{\max} - {d(k)}} \right)}}{{v(k)} \geq {\Gamma_{\min}\left( {ɛ_{\min} - {d(k)}} \right)}}{{{x(k)} \in \chi},{{u(k)} \in }}{{{x(0)} = {x(t)}},}} & (15)\end{matrix}$

wherein x(k+i) is the predicted state value at time t+i dh, x(t+i dh).

At any time t of the control one embodiment solves the problem (15) onthe future interval [t, t+N dh] and a first control input u(0) from thesequence of control inputs specifying the control action for a next timestep of control is applied during [t, t+dh] then the new state x(t+dh)is read and a new problem is solved. Problem (15) is not guaranteed tobe feasible. However, some embodiments modify the constraints toguarantee the feasibility.

The set of the feasible states

_(f) is the set that includes all the values for the state x satisfyingthe Equations (2), (6), (7). The control invariant subset C_(x) of theset of feasible states used by some embodiments is control invariantwith respect to dynamics (14) and constraints (2), (6), (7) that is, ifxεC_(x), there exists a value Uε

such that f_(d)(x,u,p)εC_(x).

Some embodiments also determine a control invariant admissible inputset, such that any control input of the control invariant admissibleinput set applied to any state in the control invariant set maintainsthe state of the train within the control invariant subset. The controlinvariant admissible input set in some embodiments,C_(u)(x)={uεU:f_(d)(x,u,p)εC_(x)}, is the set of inputs that can beapplied to any state in the control invariant set with the guaranteethat the updated state is in the invariant set.

Accordingly, some embodiments select a control action for the movementof the train corresponding to a control input from the control invariantadmissible input set. The modified constrained control problem is

$\begin{matrix}{{{\min \mspace{11mu} {F\left( {x(N)} \right)}} + {\sum\limits_{k = 0}^{N}{L\left( {{x(k)},{u(k)}} \right)}}}{{x\left( {k + 1} \right)} = {f_{d}\left( {{x(k)},{u(k)},p} \right)}}{{y(k)} = {\begin{bmatrix}{d(k)} \\{v(k)}\end{bmatrix} = {h_{d}\left( {x(k)} \right)}}}{{{x(k)} \in C_{x}},{{u(k)} \in {C_{u}\left( {x(k)} \right)}}}} & (16) \\{{x(0)} = {{x(t)}.}} & \;\end{matrix}$

If x(t)εC_(x) then the modified problem is feasible, and when the inputu is applied to the train, the problem generated at the next time stept+dh is going to be feasible because x(t+dh)εC_(x). Thus, if the firstproblem generated when the controller is initialized is feasible, thegenerated trajectory always satisfies constraints (2) and hence thetrain stops where required.

Robust Control Invariant for Uncertain Parameters

In some cases the value of the variables in the parameter vector p in(5) is not exactly known. For instance, only an upper and lower boundmay be known, or more generally that the parameter vector p has one ofthe values in a set P, which may also be constantly changing within thisset.

It is realized that the control strategy can be modified to guaranteeprecise stopping in the presence of constraints by ensuring that theconstraints in (2), (6), (7) are satisfied at any time instant for allvalue of the parameter vector. For example, some embodiments determinethe control invariant subset for a set of possible parameters of thetrain, such that for each state within the control invariant subsetthere is at least one control action maintaining the state of the trainwithin the control invariant subset for all parameters from the set ofpossible parameters of the train.

To this end in place of the sets C_(x), C_(u)(x) in (16) someembodiments use the sets {tilde over (C)}_(x)(P), {tilde over(C)}_(u)(x,P). The set {tilde over (C)}_(x)(P) is a subset of X_(f) suchthat for all states x that are in {tilde over (C)}_(x)(P), there existsan input u such that f_(d)(x,u,p)ε{tilde over (C)}_(x)(P), for all thevalues p in P. The set {tilde over (C)}_(u)(x,P), which is the set ofinputs that can be applied to state x in {tilde over (C)}_(x)(P), suchthat f_(d)(x,u,p)ε{tilde over (C)}_(x)(P) is in {tilde over (C)}_(x)(P)for all p in P.

Thus the problem for stopping the train with uncertain parameter valuesis

$\begin{matrix}{{{{\min \mspace{11mu} {F\left( {x(N)} \right)}} + {\sum\limits_{k = 0}^{N}{L\left( {{x(k)},{u(k)}} \right)}}}{{x\left( {k + 1} \right)} = {f_{d}\left( {{x(k)},{u(k)},\hat{p}} \right)}}{{y(k)} = {\begin{bmatrix}{d(k)} \\{v(k)}\end{bmatrix} = {h_{d}\left( {x(k)} \right)}}}{{x(k)} \in {{\overset{\sim}{C}}_{x}(P)}},{{u(k)} \in {{\overset{\sim}{C}}_{u}\left( {{x(k)},P} \right)}}}{{{x(0)} = {x(t)}},}} & (17)\end{matrix}$

where {circumflex over (p)}εP may not be the actual value of theparameters, which is unknown.

If x(t)ε{tilde over (C)}_(x)(P) then the modified problem is feasible,and when the input u is applied to the train, the problem generated atthe next time step t+dh is also feasible because x(t+dh)ε{tilde over(C)}_(x) (P) for all real values of p in P. Thus, if the first problemgenerated when the controller is initialized is feasible, the generatedtrajectory always satisfies constraints in (2), (6), (7) and hence thetrain stops where required

FIG. 7B shows the relation between different feasible regions. Then thefeasible regions 101 includes the control invariant set 401, which inturn include the control invariant set 402 for a set of possibleparameters of the train.

Control Invariant Set Computation

FIG. 8 shows a block diagram of a backward-reachable region computationfor determining the control invariant subset starting from the feasibleregion. The backward-reachable region computation determines the sets{tilde over (C)}_(x)(P), {tilde over (C)}_(u)(x,P) for uncertainty setP. The sets C_(x), C_(u)(x), can be generated by the same computationwhere the set P includes only a single value.

The backward-reachable region computation initializes 601 a current set

_(c) to the feasible set

_(f) and determines 602 a previous set of states

_(p) as a subset of the current set

_(c) such that for all states x in

_(p) there exists an input u in

such that for all the possible values of the parameters p in P, theupdated state lies in the current set

_(c).

If 603 the previous set

_(p) is empty, it is not possible 604 to guarantee feasibility ofproblem (17), which means that the set P where the parameters should bereduced in size. If 605 the current set and the previous set are equal,that is also 606 the set {tilde over (C)}_(x)(P) otherwise, the previousset is assigned 607 to be the current set and the computation iterates608 again. When the set {tilde over (C)}_(x)(P) is found the lastlycomputed set of state-input couples is the control invariant admissibleinput set {tilde over (C)}_(u)(x,P) for all xε{tilde over (C)}_(x) (P).

FIG. 9 shows a block diagram of an exemplar implementation ofdetermining 602 the previous set of states according to one embodiment.The embodiment identifies 701 the state-input couple that generates anupdated state that is in the current set for all the values of theparameters, and projects 702 the state input couples into state values,i.e., the embodiment identifies the states that belongs to at least oneof such state-input couples.

When the stopping constraint are define by the constraints in Equation(2), the computations of step 701 can be further simplified In thiscase, the sets

and

are described by linear inequalities, and a set of linear modelsdescribed by matrices A_(i), B_(i), i=1, . . . l and B_(w), anddisturbance set co({w_(j)}_(j=1) ^(η)) and can be found such that forall x in

, u in

f(x,u,p)εco({A _(i) x(k)+B _(i) u(k)}_(i=1) ^(l))⊕B _(w) co({w_(j)}_(j=1) ^(η))  (18)

for all p in P, where “co” denotes the convex hull and ⊕ denotes the setsum.

The linear models in (18) can be computed for instance by taking themaximum and minimum of the parameters that form vector p allowed by P,and/or of their combinations. Equations (18) convers also the case whenall the parameters are perfectly known, since in that case only onemodel is used l=1, η=1.

FIG. 10 shows a block diagram of a method for computation of the couplesof states-inputs 701 when Equations (18), (2) and linear inequalitiesdescribe the sets

and

. The method considers the current set as

H ^((c)) x≦K ^((c))  (19)

and determines 801 the worst case effects of the additive disturbance won the current set,

$\begin{matrix}{{\lbrack S\rbrack_{i} = {\max\limits_{w \in {{co}{({\{ w_{j}\}}_{j = 1}^{\eta})}}}\left\lbrack {H^{(c)}B_{w}w} \right\rbrack_{i}}},{i = 1},\ldots \mspace{14mu},n_{q},} & (20)\end{matrix}$

Next, the method reduces 802 the current set by the worst casedisturbance effects to produce a reduced current set

_(s) described by

H ^((c)) x≧K ^((c)) −S _(i),  (21)

and then determines 803 the couples (x, u) such that the updated stateis inside the current set for all the vertex systems in (18), i.e.,A_(i)x+B_(i)uε

_(s), ∀i=1, . . . l.

Train Stopping Control Systems based on Control Invariant Sets and SoftLanding Constraints

FIG. 11 shows a diagram of a predictive control system according tonsome embodiments of the invention. The control system 1101 has a modelof the train dynamics 1102 such as a model (14), the equations 1103 ofthe train constraints (6), (7), and the control invariant sets 1104. Thecontroller receives information from train on board sensors 1105, suchas wheel speed sensors, electric motor current, braking systemspressure, and possibly from external sensors 1106, such as GPSsatellites, base stations, sensors along or within the rail tracks.

Based on such information the controller selects commands for thepropulsion force needed to influence the train motion which are sent tothe train 1108 and used in the propulsion system, where a positive forceis actuated by the traction motors, and a negative force is actuatedfrom the braking system. The controller may solve the problems (12) or(13) from current time T to t_(f)=∞, thus obtaining full trajectory forthe input that is sent to the train propulsion system. More commonly,the controller operates in a receding horizon strategy as described inFIG. 601 thus receiving data from sensors that amounts to acquiring thecurrent state 501, initializing 502 and then solving 503 a finite timeoptimal control problem either (15), or (16), or (17), and commanding504 the first component of the computed input to the train propulsionsystem.

If the constrained control of Equations (12) or (13) or (15) or (16) or(17) is solved always with a feasible solution, then the train stops inthe desired range of locations. Furthermore, for the control describedin Equations (16) and (17) guarantees that if the first problem solvedwhen the control system is first activated is feasible, all thesubsequent problems are feasible, and hence the train stops in thedesired range of locations. It is also realized that in order for thefirst problem to be feasible, it is enough to initialize the controllerwhen the current state x(t) of the train system is in the controlinvariant set, x(t)εC_(x) for (16) and x(t)ε{tilde over (C)}_(x)(P) for(17).

Furthermore, it is realized that by using the control invariant subsetdetermined using the backward-reachable region computation starting fromthe feasible region, the train control system does not require acalibration to achieve the primary target, because the control invariantsubset is determined independently of all the controller calibrationparameters, such as the length of the horizon, h, and the cost functioncomponents L, F.

These parameters can be selected to obtain secondary objectives of thecontroller such as minimum time stopping, for which L are selected as

L=d ²,  (22)

minimum braking effort

L=F ²,  (23)

which also provides smooth deceleration, minimum velocity stopping

L=v ²,  (24)

minimum energy

L=u _(v) ²  (25)

which penalizes only the use of traction motors by defining u_(v)≧F, ora combination of the above functions. For (22), (24), F=L, for (23),(25) F=0. The horizon length h can be selected based on timingrequirements since longer horizon provides better performance withrespect to the select secondary objective, but requires longercomputations for the controller to generate the commands.

In the embodiment using the dynamics on the right hand side of (18) areused, and the stopping constraints includes linear inequalities, theproblems (15), (16), (17) can be converted into quadratic programmingproblems that can be solved more effectively.

FIG. 12 show a block diagram of a controller that does not use a fullmodel of the train dynamics and constraints according to an alternativeembodiment of the invention. This embodiment uses only the controlinvariant admissible input set C_(u)(x) or {tilde over (C)}_(u)(x,P),and a library of predefined control functions. For example, thisembodiment determines a set of control actions moving the train from acurrent state to next states, and selects the control action from theset of control actions, if a next state corresponding to the controlaction is the control invariant subset.

For example, the embodiment acquires 1201 the train states from sensors1105, 1106, then it selects 1202 one of the available control functionsand evaluate the corresponding candidate control command 1203. Then, theembodiment checks 1204, if the control action u is in the controlinvariant admissible input set for the current state x, that isuεC_(u)(x) or uε{tilde over (C)}_(u)(x,P), and if so, that controlaction is applied 1205. Otherwise 1206, the embodiment checks if morecandidate functions are available. If not 1207 an error is returned,otherwise another candidate function selection is operated. Becauseselecting a control action in the control invariant sets guarantees thatthe state remains in the control invariant set, unless an error isreturned, the train stops in the desired stopping range.

The above-described embodiments of the present invention can beimplemented in any of numerous ways. For example, the embodiments may beimplemented using hardware, software or a combination thereof. Whenimplemented in software, the software code can be executed on anysuitable processor or collection of processors, whether provided in asingle computer or distributed among multiple computers. Such processorsmay be implemented as integrated circuits, with one or more processorsin an integrated circuit component. Though, a processor may beimplemented using circuitry in any suitable format.

Also, the various methods or processes outlined herein may be coded assoftware that is executable on one or more processors that employ anyone of a variety of operating systems or platforms. Additionally, suchsoftware may be written using any of a number of suitable programminglanguages and/or programming or scripting tools, and also may becompiled as executable machine language code or intermediate code thatis executed on a framework or virtual machine Typically thefunctionality of the program modules may be combined or distributed asdesired in various embodiments.

Also, the embodiments of the invention may be embodied as a method, ofwhich an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts concurrently, eventhough shown as sequential acts in illustrative embodiments.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

We claim:
 1. A method for controlling of a movement of a train to a stopat a stopping position between a first position and a second position,comprising: determining constraints of a velocity of the train withrespect to a position of the train forming a feasible area for a stateof the train during the movement, such that an upper curve bounding thefeasible area has a zero velocity only at the second position, and alower curve bounding the feasible region has a zero velocity only at thefirst position; and controlling the movement of the train subject to theconstraints, the steps of the method are performed by a processor. 2.The method of claim 1, wherein the feasible area includes a currentposition of the train and a current velocity of the train, and whereinthe first and the second positions define a stopping range for thetrain, such that the first position is closer to the current position ofthe train than the second position.
 3. The method of claim 1, whereinthe constraints are determined without having a predetermined run-curveleading the train from the current position to the stopping position. 4.The method of claim 1, wherein the controlling comprises: determining,for each time step of control, a control action moving the train from acurrent position to a next position within the feasible region, suchthat the controlling includes determining a sequence of control actionsforming an ad-hoc run-curve leading the train from the current positionto the stopping position.
 5. The method of claim 4, further comprising:determining a current state of the movement of the train; optimizing themovement of the train from the current state to subsequent states over afinite horizon of time to produce a sequence of control inputs, whereinthe optimizing is subject to the constraints; and selecting a firstcontrol input from the a sequence of control inputs specifying thecontrol action for a next time step of control.
 6. The method of claim5, wherein the movement of the train is represented by a cost function,such that the optimizing of the movement includes optimizing the costfunction, and wherein the cost function includes one or combination ofan energy required by the train to reach the stopping position, a timerequired by the train to reach the stopping position, a change of avelocity required by the train to reach the stopping position, a brakingforce required by the train to reach the stopping position, or adeceleration required by the train to reach the stopping position. 7.The method of claim 1, wherein the upper curve is a first line with afirst slope, and the lower curve is a second line with a second slopeand the first slope is greater than the second slope.
 8. The method ofclaim 7, wherein the first slope is selected based on a minimal stoppingtime, and the second slope is selected based on a maximal stopping time.9. The method of claim 1, wherein the controlling comprises: selecting acontrol action maintaining the state of the train within a controlinvariant subset of the feasible region, wherein for each state withinthe control invariant subset there is at least one control actionmaintaining the state of the train within the control invariant subset.10. The method of claim 9, further comprising: determining the controlinvariant subset using a backward-reachable region computation startingfrom the feasible region.
 11. The method of claim 10, furthercomprising: determining a control invariant admissible input set, suchthat any control input of the control invariant admissible input setapplied to any state in the control invariant set maintains the state ofthe train within the control invariant subset; and selecting a controlaction for the movement of the train corresponding to a control inputfrom the control invariant admissible input set.
 12. The method of claim10, wherein the determining the control invariant subset comprises:determining the control invariant subset for a set of possibleparameters of the train, such that for each state within the controlinvariant subset there is at least one control action maintaining thestate of the train within the control invariant subset for allparameters from the set of possible parameters of the train.
 13. Themethod of claim 12, wherein the constraints are linear inequalities andtrain dynamics are represented as a set of linear models subject toadditive disturbances wherein, the backward-reachable region computationuses the train dynamics and includes: determining a worst case effect ofan additive disturbance; and determining the backward-reachable regionas an intersection for backward reachable regions of the linear modelsin the set.
 14. The method of claim 13, wherein the linear models andthe additive disturbance are such that the state of the train, controlinputs, and the train dynamics are within a convex combination of thelinear models and values of additive disturbance for any values ofparameters of the train.
 15. The method of claim 12, wherein theconstraints are linear inequalities, such that the train dynamics arerepresented as a set of linear models subject to additive disturbances,and wherein the optimizing is obtained by a constrained quadraticprogramming.
 16. The method of claim 9, further comprising: determininga set of control functions moving the train from a current state to nextstates; and selecting the control action provided by the set of thecontrol functions, if a next state corresponding to the control actionis in the control invariant subset.
 17. A system for controlling of amovement of a train to a stop at a stopping position between a firstposition and a second position, comprising: a constraint generating unitfor determining constraints of a velocity of the train with respect to aposition of the train forming a feasible area for a state of the trainduring the movement, such that an upper curve bounding the feasible areahas a zero velocity only at the second position, and a lower curvebounding the feasible region has a zero velocity only at the firstposition, wherein the upper curve is a first line with a first slope,and the lower curve is a second line with a second slope and the firstslope is greater than the second slope; and a controller for controllingthe movement of the train subject to the constraints.
 18. The system ofclaim 17, further comprising: a control invariant subset generating unitfor determining a control invariant subset of the feasible region,wherein for each state within the control invariant subset there is atleast one control action maintaining the state of the train within thecontrol invariant subset, wherein the controller controls the movementof the train by selecting a control action maintaining the state of thetrain within the control invariant subset of the feasible region. 19.The system of claim 18, wherein the control invariant subset generatingunit determines the control invariant subset using a backward-reachableregion computation starting from the feasible region.
 20. A method forcontrolling of a movement of a train to a stop at a stopping positionbetween a first position and a second position, comprising: determiningconstraints of a velocity of the train with respect to a position of thetrain forming a feasible area for a state of the train during themovement, such that an upper curve bounding the feasible area has a zerovelocity only at the second position, and a lower curve bounding thefeasible region has a zero velocity only at the first position;determining a control invariant subset of the feasible region, whereinfor each state within the control invariant subset there is at least onecontrol action maintaining the state of the train within the controlinvariant subset; and controlling the movement of the train subject tothe constraints by selecting a control action maintaining the state ofthe train within the control invariant subset of the feasible region,wherein the steps of the method are performed by a processor.